Derivative Behaviors (f, f', f")

Front

What is the relationship between a function's increasing behavior and its derivative?

Back

When a function is increasing, its derivative (f') is positive.

Front

What does it mean if f''(a) is negative at a local maximum?

Back

It indicates that the function is concave down at that point.

Front

How can you determine if a function is concave upward using its derivative?

Back

A function is concave upward if its derivative (f') is increasing, which means f''(x) > 0.

Front

What does f'(x) < 0 indicate about the function f(x)?

Back

It indicates that the function f(x) is decreasing.

Front

What is the significance of f''(x) > 0 in terms of f'(x)?

Back

If f''(x) > 0, then f'(x) is increasing.

Front

What does it mean for a function to be concave down?

Back

A function is concave down if its second derivative (f'') is negative.

Front

If f'(x) is constant, what can be said about f''(x)?

Back

If f'(x) is constant, then f''(x) = 0.

Front

What is the definition of a local maximum?

Back

A local maximum is a point where the function value is higher than all nearby points.

Front

What does it mean if f'(x) changes from positive to negative at a point?

Back

It indicates that the point is a local maximum.

Front

How can you identify an inflection point using the second derivative?

Back

An inflection point occurs where f''(x) changes sign.